A consistent local linear estimator of the covariate adjusted correlation coefficient

Consider the correlation between two random variables (X,Y), both not directly observed. One only observes X˜=ϕ1(U)X+ϕ2(U) and Y˜=ψ1(U)Y+ψ2(U), where all four functions {ϕl(⋅),ψl(⋅),l=1,2} are unknown/unspecified smooth functions of an observable covariate U. We consider consistent estimation of the correlation between the unobserved variables X and Y, adjusted for the above general dual additive and multiplicative effects of U, based on the observed data (X˜,Y˜,U).